Use graphing technology to graph each of the following functions. From the graph, find the absolute maximum and absolute minimum values of the given functions on the indicated intervals.
f(x) =e^{-x}-e^{-3x} on 0\leq x \leq 10
Use graphing technology to graph each of the following functions. From the graph, find the absolute maximum and absolute minimum values of the given functions on the indicated intervals.
m(x) =(x + 2)e^{-2x} on x\in [-4, 4]
Use the algorithm for finding extreme values to determine the absolute maximum and minimum values of the functions
f(x) =e^{-x}-e^{-3x} on 0\leq x \leq 10
Determine the absolute max for m(x).
m(x) =(x + 2)e^{-2x} on x\in [-4, 4]
The squirrel population in a small self-contained forest was studied by a biologist. The biologist found that the squirrel population, P, measured in
hundreds, is a function of time, t, where t is measured in weeks. The function is P(t) = \frac{20}{1 + 3e^{-0.02t}}.
a. Determine the population at the start of the study, when t = 0.
b. The largest population the forest can sustain is represented mathematically by the limit as t \to \infty. Determine this limit.
The squirrel population in a small self-contained forest was studied by a
biologist. The biologist found that the squirrel population, P, measured in
hundreds, is a function of time, t, where t is measured in weeks. The function is P(t) = \frac{20}{1 + 3e^{-0.02t}}.
a. Determine the point of inflection.
b. Graph the function.
c. Explain the meaning of the point of inflection in terms of surreal population growth.
The net monthly profit, in dollars, from the sale of a certain item is given by the formula P(x) = 10^6[1 +(x -1)e^{-0.001x}], where x is the number of item sold.
Determine the number of items that yield the maximum profit. At full capacity the factory can produce 2000 items per month.
The net monthly profit, in dollars, from the sale of a certain item is given by the formula P(x) = 10^6[1 +(x -1)e^{-0.001x}], where x is the number of item sold.
Determine the number of items that yield the maximum profit. At full capacity the factory can produce 500 items per month.
Suppose that the monthly revenue in thousands of dollars, for the sale of x hundred units of an electronic item is given by the function R(x) = 40x^2e^{-0.4x} + 30, where the maximum capacity of the plant is 800 units. Determine the number of units to produce in order to maximize revenue.
A rumour spreads through a population in such a way that t hours after the rumour starts, the percent of people involved in passing it on is given by P(t) = 100(e^{-t} - e^{-4t}). What is the biggest percent of people involved in
spreading the rumour within the first 3 h? When does this occur?
Small countries trying to develop an industrial economy rapidly often try to achieve their objectives by importing foreign capital and technology. Statistics Canada data show that when Canada attempted this strategy from 1867 to 1967, the amount of U.S. investment in Canada increased from about
represented by the simple mathematical model C(t) = 0.015 \times 10^9 e^{0.07533t}, where t represents the number of years (starting with 1967 as zero) and C represents the total capital investment from US sources in dollars.
Small countries trying to develop an industrial economy rapidly often try to achieve their objectives by importing foreign capital and technology. Statistics Canada data show that when Canada attempted this strategy from 1867 to 1967, the amount of U.S. investment in Canada increased from about
represented by the simple mathematical model C(t) = 0.015 \times 10^9 e^{0.07533t}, where t represents the number of years (starting with 1967 as zero) and C represents the total capital investment from US sources in dollars.
Small countries trying to develop an industrial economy rapidly often try to achieve their objectives by importing foreign capital and technology. Statistics Canada data show that when Canada attempted this strategy from 1867 to 1967, the amount of U.S. investment in Canada increased from about
represented by the simple mathematical model C(t) = 0.015 \times 10^9 e^{0.07533t}, where t represents the number of years (starting with 1967 as zero) and C represents the total capital investment from US sources in dollars.
A colony of bacteria in a culture grows at a rate given by N(t) = 2^{\frac{t}{5}}, where N is the number of bacteria t minutes form the beginning. The colony is allowed to grow fro 60 min, at which time a drug is introduced to kill the bacteria. The number of bacteria killed is given by K(t) = e^{\frac{t}{3}}, where K bacteria are killed at time t minutes.
a. Determine the maximum number of bacteria present and the time at which this occurs.
b. Determine the time at which the bacteria colony is obliterated.
Lorianne is studying for two different exams. Because of the nature of the courses, the measure of study effectiveness on a scale from 0 to 10 for the first course is
E_1 = 0.6(9 + te^{-\frac{t}{20}}), while the measure for the second course
is E_2 = 0.5(10 + te^{-\frac{t}{10}}) . Lorianne is prepared to spend up to 30 h, in total, studying for the exams. The total effectiveness is given by f(t) = E_1 + E_2.
How should this time be allocated to maximize total effectiveness?
Find the absolute extrema of f(x) =x = e^{2x} on the interval x\in [-2, 2].
For f(x) = x^2e^x, determine the intervals of increase and decrease.
Determine the absolute minimum value of f(x) = x^2e^x.
Find the maximum and minimum values.
y = e^x + 2
Find the maximum and minimum values.
y = xe^x + 3
Find the maximum and minimum values. Sketch the function.
y = 2xe^{2x}
Find the maximum and minimum values. Sketch the function.
y = 3xe^{-x} + x
The profit function of a commodity is P(x) = xe^{-0.5x^2}, where x > 0. Find the maximum value of the function, if x is measured in hundreds of units and P, is measured in thousands of dollars.
a. Determine an equation for A(t) , the amount of money in the account
at any time t .
b. Find the derivative A^{\prime}(t) of the function.
c. At what rate is the amount growing at the end of two years? At what rate is it growing at the end of five years and at the end of 10 years?
d. Is the rate constant?
e. Determine the ratio of \frac{A^{\prime}(t)}{A(t)} for each value that you determined for A^{\prime}(t) .
f. What do you notice?